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Stationary maximal hypersurface existence with a horizon boundary

Statement

Under the exact class-(b) hypotheses and exclusion condition [F1], There is a family ${\hat{Σ}}_{s}$ of smooth spacelike maximal asymptotically flat hypersurfaces, $s \in R$ , with $\partial {\hat{Σ}}_{s} = S$ , ${\hat{Σ}}_{s}$ diffeomorphic to $Σ$ , $ϕ_{t} ({\hat{Σ}}_{s}) = {\hat{Σ}}_{s + t}$ and interiors covering $U$ and Cauchy there in the source convention. If $X$ is timelike on $U$ , or if $Ric (Z, Z) \geq 0$ for every timelike $Z$ , their interiors foliate $U$ . The source's asymptotic flatness means end graphs obeying Definition 2.4: $∣ u ∣ + \sum_{j = 1}^{k + 1} r^{j} ∣ \partial^{j} u ∣ \leq C w_{α} (r)$ , with $w_{α}$ as in the graph-translation lemma, together with the corresponding weighted Hölder condition on the highest derivatives. In particular at $α = 1$ logarithmic height growth is allowed.

Facts & Assumptions

[F1]

The complete class-(b) hypotheses and exclusion condition are as follows. Let $(M, g)$ be a smooth connected time-oriented strongly causal spacetime and $Σ$ a connected closed acausal hypersurface with boundary $S$ . Closed means closed as a subset of $M$ ; acausal means that no causal curve connects distinct points of $Σ$ . Suppose $Σ$ is the union of finitely many exterior ends $Σ_{i}$ and a compact interior $Σ_{i n t}$ with boundary, glued along their boundary components. Each $Σ_{i}$ is diffeomorphic to $R^{3} ∖ \overline{B} (R_{i})$ . For some $k \geq 2$ , $0 < α \leq 1$ , $0 < λ \leq 1$ and end-dependent $C$ , each has a neighborhood chart with $r = ∣ x ∣$ , $g_{00} < - C^{- 1}$ , $∣ g_{μ ν} - η_{μ ν} ∣ \leq C r^{- α}$ , $g_{i j} v^{i} v^{j} \geq C^{- 1} ∣ v ∣^{2}$ and $∣ \partial^{j} g_{μ ν} ∣ \leq C r^{- α - j}$ for $1 \leq j \leq k$ . Order- $k$ derivatives have difference bound $C r^{- α - k - λ} (∣ t - τ ∣^{λ} + ∣ x - y ∣^{λ})$ for $∣ t - τ ∣ \leq 1$ and $∣ x - y ∣ \leq r (x) / 2$ . The sphere bounding each end is identified diffeomorphically with a connected boundary component of its complementary portion. A Killing field $X$ is complete on $M$ , uniformly timelike on each end, tangent to $S$ , and in each chart is $(A_{i} + O (r^{- α})) \partial_{t} + O (r^{- α}) \partial_{x^{i}}$ with $A_{i} \neq 0$ . These are Definitions 2.1 and 2.3, called class (b); the definition does not require $X = 0$ on $S$ or a bifurcate Killing horizon. Set $M_{i} = ⋃_{t \in R} ϕ_{t} (Σ_{i})$ , $M_{e x t} = ⋃_{i} M_{i}$ , $B = M ∖ I_{M}^{-} (M_{e x t})$ , $W = M ∖ I_{M}^{+} (M_{e x t})$ . Let $D (Σ)$ use the source's inextendible-timelike-curve convention and put $U = int D (Σ)$ . Require $(B \cup W) \cap U = \emptyset$ .

[F2]

The declared definitions supply stationary Lorentzian geometry, ambient chronological relations, spacelike maximality and differentiated end bounds. Stationary Lorentzian data and tensor conventions Causal exterior, chronological boundaries and Cauchy data Spacelike sections, maximality and lapse–shift data Differentiated asymptotic bounds and parameter uniformity

[F3]

The five intermediate lemmas explain the quantitative end interface, invariant compact boundary, precise exclusion test, isometry covariance and graph decay limitations. Differentiated stationary end bounds imply the weighted Hölder interface A complete stationary flow preserves its compact tangent boundary Black and white exclusion is an ambient reachability condition Isometries preserve maximality and transported boundary Stationary graph translation preserves source asymptotics

External input

Import Chruściel–Wald Theorem 4.2 under exactly [F1], with asymptotic flatness in their Definition 2.4 and source domain-of-dependence/Cauchy convention. This is an existence and covariance input; its nonlinear elliptic existence argument is not reconstructed here. The five authored local lemmas establish the indicated interfaces and consequences but are not represented as a proof of the existence theorem. For an application with the explicit additional mathematical hypothesis $Ric (g) = 0$ , one has $Ric (Z, Z) = 0 \geq 0$ for every timelike $Z$ . Thus the timelike-convergence alternative applies once the class-(b) and exclusion hypotheses have been established.

Source notes

Neither the imported existence theorem nor the local graph estimate supplies stronger differentiated decay, a zero-height end gauge, or an identification of $U$ with a separately specified exterior. Consumers must establish these additional application obligations. Strong causality here is an ambient hypothesis, not merely a property of a smaller exterior. No source bifurcation-surface hypothesis has been inserted into class (b). The source defines $D$ using inextendible timelike curves (p.8); this item retains that convention rather than silently substituting the local causal-curve formulation.

Depends on

Used by · 4

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External dependencies

Piotr T. Chruściel and Robert M. Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes · Definitions 2.1 (2.1)–(2.5), pp.8–9; 2.3, p.11; (2.13)–(2.17), pp.11–12; Definition 2.4, p.13; (4.1)–(4.3) and Theorem 4.2, pp.24–25; Remark 3, p.25

There is a family $\widehat\Sigma_s$ of smooth spacelike maximal asymptotically flat hypersurfaces, $s\in\mathbb R$, with $\partial\widehat\Sigma_s=S$, $\widehat\Sigma_s$ diffeomorphic to $\Sigma$, $\phi_t(\widehat\Sigma_s)=\widehat\Sigma_{s+t}$ and interiors covering $U$ and Cauchy there in the source convention. If $X$ is timelike on $U$, or if $\operatorname{Ric}(Z,Z)\geq0$ for every timelike $Z$, their interiors foliate $U$. The source's asymptotic flatness means end graphs obeying Definition 2.4: $|u|+\sum_{j=1}^{k+1}r^j|\partial^ju|\leq Cw_\alpha(r)$, with $w_\alpha$ as in the graph-translation lemma, together with the corresponding weighted Hölder condition on the highest derivatives. In particular at $\alpha=1$ logarithmic height growth is allowed.

Hypotheses: Let $(M,g)$ be a smooth connected time-oriented strongly causal spacetime and $\Sigma$ a connected closed acausal hypersurface with boundary $S$. Closed means closed as a subset of $M$; acausal means that no causal curve connects distinct points of $\Sigma$. Suppose $\Sigma$ is the union of finitely many exterior ends $\Sigma_i$ and a compact interior $\Sigma_{\rm int}$ with boundary, glued along their boundary components. Each $\Sigma_i$ is diffeomorphic to $\mathbb R^3\setminus\overline B(R_i)$. For some $k\geq2$, $0<\alpha\leq1$, $0<\lambda\leq1$ and end-dependent $C$, each has a neighborhood chart with $r=|x|$, $g_{00}<-C^{-1}$, $|g_{\mu\nu}-\eta_{\mu\nu}|\leq Cr^{-\alpha}$, $g_{ij}v^iv^j\geq C^{-1}|v|^2$ and $|\partial^j g_{\mu\nu}|\leq Cr^{-\alpha-j}$ for $1\leq j\leq k$. Order-$k$ derivatives have difference bound $Cr^{-\alpha-k-\lambda}(|t-\tau|^\lambda+|x-y|^\lambda)$ for $|t-\tau|\leq1$ and $|x-y|\leq r(x)/2$. The sphere bounding each end is identified diffeomorphically with a connected boundary component of its complementary portion. A Killing field $X$ is complete on $M$, uniformly timelike on each end, tangent to $S$, and in each chart is $(A_i+O(r^{-\alpha}))\partial_t+O(r^{-\alpha})\partial_{x^i}$ with $A_i\ne0$. These are Definitions 2.1 and 2.3, called class (b); the definition does not require $X=0$ on $S$ or a bifurcate Killing horizon. Set $M_i=\bigcup_{t\in\mathbb R}\phi_t(\Sigma_i)$, $M_{\rm ext}=\bigcup_iM_i$, $B=M\setminus I_M^-(M_{\rm ext})$, $W=M\setminus I_M^+(M_{\rm ext})$. Let $D(\Sigma)$ use the source's inextendible-timelike-curve convention and put $U=\operatorname{int}D(\Sigma)$. Require $(B\cup W)\cap U=\varnothing$.

Reading limits: Consulted the PDF text of gr-qc/9304009: section 2 introductory conventions and Definitions 2.1, 2.3 and 2.4 (printed pp.8–13), equations (2.1)–(2.5) and (2.13)–(2.17), and section 4 equations (4.1)–(4.3), Theorem 4.2 and Remark 3 (printed pp.24–25). Checked the stated hypotheses, domain-of-dependence convention, boundary and covariance conclusions, conditional foliation and allowed logarithmic graph growth. The nonlinear elliptic existence proof and its cited sources were not independently reconstructed or reviewed; PDF screenshots were unavailable.

Sources

Piotr T. Chruściel and Robert M. Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes · Definitions 2.1 (2.1)–(2.5), pp.8–9; 2.3, p.11; (2.13)–(2.17), pp.11–12; Definition 2.4, p.13; (4.1)–(4.3) and Theorem 4.2, pp.24–25; Remark 3, p.25